FP2 June 2018 Q4

EdexcelOld spec7 marksInequalities

4. Use algebra to find the set of values of \(x\) for which \[\left|x^2 - 2\right| \gt 4x\] (7)

Scheme	Marks
Note: Candidates may include a sketch such as the one shown at some point in their working. Please be aware that this sketch without algebra used to find the critical values merits 0 marks. Marks may be only be awarded for the algebra used
NB First 4 marks are available with \(=\), \(\gt\) or \(\lt\) used
\(x^2 - 2 = 4x\) ① Form 3 TQ and* attempt to solve - may be implied by correct value(s) (allow decimals 4.449 ,-0.449..)*	M1
\(x = 2 \pm \sqrt{6}\) or \(2 + \sqrt{6}\) Correct exact values or value (NB: Corresponding 3TQ must have been seen)	A1
\(x^2 - 2 = -4x\) ② Form 3 TQ and* attempt to solve - may be implied by correct value(s) (allow decimals -4.449 ,0.449..))*	M1
\(x = -2 \pm \sqrt{6}\) or \(x = -2 + \sqrt{6}\) Correct exact values or value (NB: Corresponding 3TQ must have been seen)	A1
\(x \gt\) larger root of ① or \(x \lt\) larger root of ② Forms at least one of the required inequalities using their exact values Must be a strict inequality Depends on either previous M mark	dM1
One of \(x \lt -2 + \sqrt{6}\) or \(x \gt 2 + \sqrt{6}\) Or exact equivalent	A1
Both of \(x \lt -2 + \sqrt{6}\) or \(x \gt 2 + \sqrt{6}\) No others seen. Exact equivalents allowed Allow “or” or “and” but not \(\cap\) if set notation used	A1
	(7 marks)

Scheme	Marks
\(\left(x^2 - 2\right)^2 = 16x^2\) Square both sides and* attempt to solve quadratic in \(x^2\) may be implied by correct value(s) (allow decimals 19.79... -0.202..)*	M1
\(x^2 = 10 \pm \sqrt{96}\) \(x^2 = 10 \pm 4\sqrt{6}\) oe	A1
\(x = 2 \pm \sqrt{6}\) and \(x = -2 \pm \sqrt{6}\) \(\left(x = 2 + \sqrt{6}\text{ and }x = -2 + \sqrt{6}\text{ sufficient}\right)\) Valid attempt required to find exact form for \(x\) e.g. \(\left(a + \sqrt{b}\right)^2 = 10 \pm \sqrt{96}\)	M1A1
\(x \gt\) largest root or \(x \lt\) 2nd largest root As main scheme	dM1
As main scheme As main scheme	A1,A1